We recall the next property associated with the sub-Laplace operator on
\(\mathbf {H}_n\)  (one may see some detail explanations and deductions in
[1]}, [2]}):
\(\mathrm {d}\pi _{\lambda }(\Delta _{\mathrm {H}})=-|\lambda |\mathrm {H}_{w}:=-|\lambda |\left(-\Delta _{w}+|w|^2\right).\) 
